In this section you will gain a deeper understanding of the Ideal Gas Law by performing comparisons and establishing relationships between a) the Ideal Gas Law and Boyle's law, b) the Ideal Gas Law and Charles law, c) the Ideal Gas Law and Avogadro's Law, and d) kinetic energy and velocity.

Remember that the Ideal Gas Law is of the form:

PV = nRT = NkT

Exercise 1: Relationship Between Volume (V) and the Number of Particles (n)
In this exercise you will change the number of particles and collect volume data, which will be recorded in the table below. Average volume will be determined using a spreadsheet. When changing values in the applet you must hit "Enter" in order for the change to take place. Also remember to give the system time to equilibrate after each change is made.

1. Use the following initial settings: particles = 50, pressure = 20.0, velocity = 100.0. After the system equilibrates, click the mouse button to freeze the system, and record three volume readings in the table below. Change the number of particles as indicated and record the new volumes in the table.

2. Transfer the data to a spreadsheet, determine the average volume, and prepare a graph of average volume (yaxis) vs. # of particles (xaxis). If you are not familiar with spreadsheet operation, please see the tutorial Virtual Exploration of Acids and Bases - Part 1 for reference. Based on the graph, what sort of relationship exists between the volume and number of particles?

NOTE: Exercise 1 shows that the volume (V) of the gas was directly proportional to the number of particles, which is called Avogadro's Law.

Exercise 2: Relationship Between Volume (V) and Particle Velocity (v)
1. Use the following initial settings: particles = 100 pressure = 50.0 velocity = 100.0 As before, allow time for the system to equilibrate after changes are made. Change the particle velocity as indicated below, and record the data.

2. Transfer the data to a spreadsheet and determine the average volume, and prepare a graph of average volume (yaxis) vs. particle velocity (xaxis). Is there a linear relationship?

NOTE: In order to understand the relationship between volume and particle velocity, there is some additional information to take into account. Recall that objects in motion (like gas particles) have kinetic energy due to their motion given by the equation: Kinetic Energy = 0.5mv^2, where m = particle mass, and v = particle velocity. The simulation demonstrates that at a given particle velocity setting, there is actually a range of individual particle velocities. Go back to the simulation and carefully observe the particles. Some move slowly while others move quickly. The kinetic energy of the collision of the particles with the piston is what determines the volume of the gas. So we do not expect a linear relationship between volume and particle velocity, but a relationship that depends on velocity squared. The proportionality between volume and temperature was first discovered by Jacques Charles around 1787, which is called Charles's Law.

Exercise 3: Relationship Between Volume (V) and Pressure (P)
1. Now that you possess the appropriate skill set, you can proceed on your own and gather the data in the table below. Recall the ¡§equilibration time¡¨ when you change parameters in the ideal gas applet. Also, you have to hit "Enter" after parameters are changed for them to take effect. Use the following initial settings: particles = 225 pressure = 15.0 velocity = 100.0.


2. In your spreadsheet, plot average volume (yaxis) vs. pressure (xaxis). The relationship between pressure and volume is not obvious from this plot. In order to have a linear relationship between the variables, a simple function of the pressure is required. Use the spreadsheet and try some! Set up columns where you take the ln(P), (P) 0.5 , inverse (P), etc., until you observe a linear relationship: V = f(P). Once you have determined the proper function of P, place the appropriate values in the blank column in the above table.

NOTE: Exercise 3 shows that the volume (V) was inversely proportional to the pressure (P), which is Boyle's law, discovered in 1662.